A Further Decay Estimate for the Dziubański–Hernández Wavelets

Moritoh, Shinya; Tomoeda, Kyoko

doi:10.4153/cmb-2010-027-3

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…Hence we get a desired wavelet by Theorem . □ Remark In this proof we used a function

g \in G_{ξ}^{1 + 0}

, whose existence is shown by J. Dziubański and E. Hernández together with further decay estimates for g given by S. Moritoh and K. Tomoeda . We shall also show a useful concrete example of functions belong to

G_{ξ}^{1 + 0}

in Appendix.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…At the end of this section, we also introduce a critical case for Theorem B. Dziubański and Hernández and Moritoh and Tomoeda showed a wavelet ψ satisfying

\hat{ψ} \in ⋂_{s > 1} G_{ξ}^{s}

, when

ψ (x)

is band limited, i.e.,

ψ \in {scriptA}_{x}

. As for non‐Gevrey wavelet

ψ (x)

, we prove the following in Section : Theorem There exists a wavelet ψ satisfying

ψ \in C_{x}^{\infty} ∖ ⋃_{s > 1} G_{x}^{s}

and

\hat{ψ} \in ⋂_{s > 1} G_{ξ}^{s}

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the wavelets having Gevrey regularities and subexponential decays

Fukuda

Kinoshita

Uehara

2013

Mathematische Nachrichten

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“…Hence we get a desired wavelet by Theorem . □ Remark In this proof we used a function

g \in G_{ξ}^{1 + 0}

G_{ξ}^{1 + 0}

in Appendix.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…At the end of this section, we also introduce a critical case for Theorem B. Dziubański and Hernández and Moritoh and Tomoeda showed a wavelet ψ satisfying

\hat{ψ} \in ⋂_{s > 1} G_{ξ}^{s}

, when

ψ (x)

is band limited, i.e.,

ψ \in {scriptA}_{x}

. As for non‐Gevrey wavelet

ψ (x)

, we prove the following in Section : Theorem There exists a wavelet ψ satisfying

ψ \in C_{x}^{\infty} ∖ ⋃_{s > 1} G_{x}^{s}

and

\hat{ψ} \in ⋂_{s > 1} G_{ξ}^{s}

.…”

Section: Introductionmentioning

confidence: 99%

On the wavelets having Gevrey regularities and subexponential decays

Fukuda

Kinoshita

Uehara

2013

Mathematische Nachrichten

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“…In [3] Dziubański and Hernández constructed an orthonormal wavelet with subexponential decay; see also [4] for other orthonormal wavelets having that decay. Their construction has been generalized by various other authors [14,16]. The Dziubański-Hernández wavelets are of Lemarié-Meyer type (i.e., band-limited), hence, they are entire functions.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution expansions and wavelets in Gelfand–Shilov spaces

Pilipović

Rakić

Teofanov

et al. 2020

RACSAM

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We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziubański-Hernández construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand-Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.2010 Mathematics Subject Classification. 42C40, 46E10, 46F05, 46F12.

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Band-limited wavelets beyond Gevrey regularity

Teofanov,

Tomić,

Tutić

2024

Int. J. Wavelets Multiresolut Inf. Process.

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It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert [Formula: see text]-function.

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A Further Decay Estimate for the Dziubański–Hernández Wavelets

Abstract: We give a further decay estimate for the Dziubański–Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley–Wiener theorem for ultradifferentiable functions.

Cited by 3 publications

References 7 publications

On the wavelets having Gevrey regularities and subexponential decays

On the wavelets having Gevrey regularities and subexponential decays

Multiresolution expansions and wavelets in Gelfand–Shilov spaces

Band-limited wavelets beyond Gevrey regularity

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